Theorem 0nnq 10965

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5718	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10953	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4081	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3978	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 197	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2107  ∀wral 3060  ∅c0 4332   class class class wbr 5142   × cxp 5682  ‘cfv 6560  2^nd c2nd 8014  Ncnpi 10885   <_N clti 10888   ~_Q ceq 10892  Qcnq 10893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205  df-xp 5690  df-nq 10953

This theorem is referenced by:  adderpq  10997  mulerpq  10998  addassnq  10999  mulassnq  11000  distrnq  11002  recmulnq  11005  recclnq  11007  ltanq  11012  ltmnq  11013  ltexnq  11016  nsmallnq  11018  ltbtwnnq  11019  ltrnq  11020  prlem934  11074  ltaddpr  11075  ltexprlem2  11078  ltexprlem3  11079  ltexprlem4  11080  ltexprlem6  11082  ltexprlem7  11083  prlem936  11088  reclem2pr  11089